who have had questions given them, and the pupil named arises, repeats, and analyzes the problem assigned him.

This method is good discipline for the memory.

Fourth method.

Two pupils are designated as chiefs, and choose alternately from among the other members of the class such as they deem the best scholars in mental arithmetic for a trial of skill. The teacher gives out problems alternately and marks the failures or the number of questions which each side solves correctly; and at the close of the lesson gives the result. Or he causes each pupil that fails to take his seat, and the side that has the largest number of pupils standing at the close of the lesson is pronounced the best.'

Motivation. The plan for securing motive which is offered by the writers on the teaching of arithmetic, whom we have quoted on the preceding pages, is by appeal to artificial incentives. The pupil attended to the example in addition because, if he did not, he knew his failure to attend would be immediately discovered by both his classmates and the teacher. Being thus caught in the act, the penalty followed, a lowering of his rank in class, a reprimand by the teacher, or a severe punishment. Or the pupil wished to secure the approbation of his teacher or parents. Knowing the shortcomings of his classmates he attended in order that he might profit by their failures. As soon as the pupil reciting faltered or made an error he was ready to take up the solution of the problem and receive his reward in the approval of the teacher or in the anticipation of the reception which would be given at home to his report card. Or perhaps his reward came from the superior position which he had attained in the class. In putting one division of the class against the other the instinct of emulation was appealed to. Or where the class was small the contest was between the individual pupils. There was an appeal to the pupil's pride when he knew his work was to be examined and marked by another pupil.

Motive was secured in other ways. The puzzle type of problem stimulated the pupil's curiosity. Some problems were practical. The primary work and the rapid drill were immediately interesting to many. But these ways of securing motive were, for the most part, used unconsciously. When a teacher wrote of how attention. was secured these phases of motive were mentioned only incidentally or not at all. Occasionally, but usually in respect to other school subjects, motive by conflict of ideas was mentioned. David P. Page, in a text on Theory and Practice of Teaching, gives a list of good incentives. They are: (1) Desire of the approval of parents and teachers; (2) desire of advancement; (3) desire to be useful; (4) desire to do right; (5) natural love in the child for acquisition and a natural desire to know.

1 P. 25.

The idea and practice of securing motive in this period was characterized by there being no intrinsic relation between the purpose which ne pupil recognized and the subject matter studied.

Problems solved according to a formula.-In mental arithmetic, which was considered to be especially suitable for developing the reasoning, the solution was accomplished by applying a syllogistic ormula. Ray says in Hints to Teachers, Intellectual Arithmetic (copyright 1860): .

A method of solving questions in mental arithmetic now much used is the following, called the "Four-step method:"

Illustrations. First step, James gave 7 cents for apples and 8 cents for peaches; how many cents did he spend? Second step, as many as the sum of 7 and 8 cents. Third step, 7 cents and 8 cents are 15 cents. Fourth step, hence, if James gave 7 cents for apples and 8 cents for peaches he spent 15 cents.

Again: First step, 4-fifths of 25 are how many times 6? Second step, as many times 6 as 6 is contained times in 4-fifths of 25. Third step, 1-fifth of 25 is 5, 4-fifths are 4 times 5, which are 20; 6 in 20 is contained 3 and 2-sixths times. Fourth step, therefore, 4-fifths of 25 are 3 and 2-sixths times 6.

Some writers insisted that these forms of analysis were to be committed to memory. In the following quotation the author believes that a verbatim memorizing of the forms of analysis will make the pupils all the better reasoners:

After the pupils are familiar with the process and have received sufficient drill they should be taught to analyze problems. The teacher should see that the analysis is thoroughly understood and accurately recited. They should be required to write out an analysis, and the pupil that presents the most simple and concise analysis should write it on the board, subject to the criticism of the class. See that the language is used correctly; that it tells the "truth, the whole truth, and nothing but the truth." Now require every member of the class to commit the analysis verbatim as he would a demonstration in Euclid-for experience teaches that those pupils who are critically close in committing verbatim the demonstrations in geometry make by far more accurate reasoners and ready mathematicians.1

When the pupil was furnished with a stereotyped form for the solution of every problem all opportunity for reasoning was eliminated except such as there might be in identifying the particular problem with the appropriate formula. Therefore the types of problems were mixed, to form promiscuous and miscellaneous lists of problems. Brooks says:

It will be frequently noticed that, after beginning the lesson with the typical problem, variations are made both in the conditions of the question and in their application to other objects than those named in the original problem. This is done to give variety to the exercises and to afford discipline to the pupil.

Assisting the pupil.-The assistance which the teacher rendered the pupil consisted mainly in holding him to certain fixed standards, in drilling upon what was considered fundamental, and in explaining difficult operations and problems. Little effort was made to assist the pupil to think. Developing a process or topic was not consciously

1 De Graff, The Schoolroom Guide, 1877.

attempted. The explanations by the teacher were simply told to the pupil. Whether the pupil understood the explanation or not, he was expected to remember it. If the difficulty was sufficiently important, the pupil was drilled upon the manner of overcoming it. In this way the learning was largely by conscious imitation, with sufficient repetition by the pupil to fix the subject matter in the mind. It seems to have been recognized that expression assisted in making the impression. The prominence given to the explanations in class by pupils was in part due to this belief.

Deductive method. We have seen how the texts of this period became deductive in form after 1857. The instruction followed the texts closely. The "rule" was again emphasized. If the pupil was able to subsume a problem under a known rule, the rule could take care of the answer. A report of the investigation of schools in Connecticut, in 1887-88, contains the following comment upon the attitude of the pupil toward the rule:

The method in arithmetic is illustrated by the course which most children will take after long instruction in such schools. If they are given a problem of one or two steps, they will first see what rule it comes under. If it does not come under any rule with which they are familiar, they will take a book and see if they can find an example like it. If they fail in this search, they then begin to cipher at random, multiplying and dividing in the hope that it may turn out right.'

The rules were not developed as Colburn did in his texts. The pupil was scarcely allowed to make a hypothesis when a new type of problem was reached, and to work out a solution of his own. Instead the rule was given to him ready made.

Objective teaching.-The use of objective materials, beans, grains of corn, pieces of crayon, etc., is recommended by Ray for the younger pupils. He also describes what he terms "arithmometer," an instrument for representing objectively the number facts of the four operations. However, he cautions against "frequent use of artificial aids," for it "tends to prevent the pupil from exercising his own intellectual powers, and thus, if carried too far, is productive of positive injury." In an edition of Greenleaf's primary book he says:

The First Lessons in Numbers has been prepared in the belief that the objective presentation of numbers is best suited to the comprehension of the child. The teacher who uses this book is expected to make constant use of counters, blocks, or other visible objects, that from the outset the child may have correct ideas of numbers. The copious illustrations found throughout the book are intended as aids in this direction. One writer on the teaching of arithmetic (1877) says: "Construct the addition tables at first by the use of objects." He advises the same plan for multiplication. Illustrations in the form of cuts became a feature of the primary arithmetics, but I have found none as profusely illustrated as Emerson's Part First. A few illustrations are found in the "mental" and the "practical" arithmetics, but usually only for elucidating a topic of peculiar difficulty.

1 Connecticut School Documents, No. VII, p. 224.

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The second Pestalozzian movement in the United States, usually known as the Oswego movement, emphasized almost exclusively objective teaching. This movement, which dates from 1860, appears to have had but little direct influence upon the teaching of arithmetic. There was only a slight increase in the use of objective materials in arithmetic after 1860.

Drill. We have already shown that skill and thoroughness were emphasized as ends to be attained. They were to be secured by drill. These goals of instruction were given increased importance in the latter part of this period. Drill devices and drill cards were given by a number of authors. Drill to make certain parts of the subject matter mechanical was insisted upon. De Graff, in The Schoolroom Guide, 1877, says:

The teacher should see that the tables are thoroughly committed to memory by requiring pupils to recite them backward and forward regularly and irregularly. Excite emulation among members of the class in regard to the mechanical execution of the work, because careless habits formed will ever be a source of annoyance to both teacher and pupil.'

In Felter's Primary Arithmetic the teacher is advised:

In order to secure thoroughness, give short lessons and spend much time in daily review. If in the exercise of "fours," do not proceed until everything that precedes is as familiar as the alphabet. If it required one month, take it; if one year, the time can not be better spent. Never allow a pupil who, habitually, misses over 10 per cent of the given exercise to remain in the class.

Some reports upon the teaching of arithmetic.-In 1887-88 a committee reported upon the condition of schools in New London County, Conn. The report was based upon tests and visitation. In respect to arithmetic they say in part:

Perhaps a half-dozen schools taught elementary arithmetic by systematically developing number, but almost invariably the teacher answered the question as to what method was used by the stereotyped phrase "follow the book," and this was literally true not only for elementary but for advanced classes.

Months and terms are spent in counting, learning to write unheard-of numbers, bawling the multiplication and perhaps other tables. No systematic development of number is thought of. No concrete examples, except the few in a small book are given. No thorough drill is attempted. No rapid handling of numbers, no accuracy with figures, no training of the reason, is the result. Most so-called mental examples have been carefully studied before the recitation. Definition and rules will be repeated fluently, and yet the pupil is unable to perform simple examples involving one or two steps of reasoning. One or two illustrations are pertinent.

A boy over 10 years of age was being taught to count one hundred, but could not tell the sum of two and two. The teacher gave as the reason for teaching him thus to count, before he could add, that "when he received change at the store he could count it."

In another school, a class of three gave with great fluency the definition of "units," "arithmetic," "counting," "scale," "counting off," "group," etc. They read numbers up to sextillions, but could not tell how many fours there were in 16. The teacher said that they had never done anything in multiplication or division. These

1 P. 189.

children had been in school about four years. It is not to be wondered at, then, that under such unnatural methods so many children attend school seven and eight years without reaching percentage and its applications to interest.1

A more elaborate investigation was made of the schools of New Haven County, Conn., in 1890-91. The committee examined 167 districts. They sum up their opinions with respect to arithmetic in this sentence:

Arithmetic has thus become a science of difficult trifles and intricate fooleries peculiar to common schools, and remarkable chiefly for sterility and ill-adaptedness to any useful purpose.

In another place they give a description of the actual activities of the schoolroom which is illuminating not only as to content, but to aim and method as well:

In many districts the main thing in arithmetic is the definitions. In one school the first class was questioned as follows: "Spell arithmetic." "What is arithmetic?" "What is Roman notation?" "What is a figure?" and so on during the recitation periods. The definitions must be word for word as in the book.

The answer to the question, "How is a fraction expressed?" was given "by writing one number above the other." This was immediately corrected by the teacher to "by placing one number above the other." 2

While these surveys were made with some care, they covered only a very limited area. For this reason it is hazardous to draw generalizations, but other evidence indicates that the conditions described in these reports are typical of much of the instruction in arithmetic at this time. However, when we compare the instruction in arithmetic during this period with that of the ciphering-book period, progress is shown. There is an enlarged concept of the function of the teacher, and if we take into account that at this time arithmetic was a "required" subject and not an "elective," it is probable that the results secured were superior.

Colburn's influence upon the teaching of arithmetic.-A comparison of Colburn's method of teaching arithmetic with the practice during this period reveals that he influenced the teaching of the subject much less than he did the texts. The use of objective materials and the oral instruction which mental arithmetic made necessary may be attributed to him.

He advocated class instruction and discussed its technique in his address on the "Teaching of arithmetic," but it is doubtful if the adoption of this plan of instruction was due to his advocacy of it. Outside of these features, which were a result of his texts rather than his presentation of the method of teaching arithmetic, there is little trace of Colburn's influence. This condition is easily explained by the fact that a method of teaching is much less tangible than the form of a textbook. Also, in the making of texts there are few persons concerned, as compared with the number of teachers.

1 Connecticut School Documents, No. VII, p. 117.

2 Idem, No. XVI, p. 29.